Mathematical Surveys and Monographs Volume 218 Geometry and Dynamics in Gromov Hyperbolic Metric Spaces With an Emphasis on Non-Proper Settings
Tushar Das David Simmons Mariusz Urbaƙski
American Mathematical Society 10.1090/surv/218
Geometry and Dynamics in Gromov Hyperbolic Metric Spaces
With an Emphasis on Non-Proper Settings
Mathematical Surveys and Monographs Volume 218
Geometry and Dynamics in Gromov Hyperbolic Metric Spaces
With an Emphasis on Non-Proper Settings
Tushar Das David Simmons Mariusz Urbaƙski
American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein
2010 Mathematics Subject Classification. Primary 20H10, 28A78, 37F35, 20F67, 20E08; Secondary 37A45, 22E65, 20M20.
For additional information and updates on this book, visit www.ams.org/bookpages/surv-218
Library of Congress Cataloging-in-Publication Data Names: Das, Tushar, 1980- | Simmons, David, 1988- | Urba´nski, Mariusz. Title: Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non- proper settings / Tushar Das, David Simmons, Mariusz Urba´nski. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Mathe- matical surveys and monographs ; volume 218 | Includes bibliographical references and index. Identifiers: LCCN 2016034629 | ISBN 9781470434656 (alk. paper) Subjects: LCSH: Geometry, Hyperbolic. | Hyperbolic spaces. | Metric spaces. | AMS: Group theory and generalizations – Other groups of matrices – Fuchsian groups and their general- izations. msc | Measure and integration – Classical measure theory – Hausdorff and packing measures. msc | Dynamical systems and ergodic theory – Complex dynamical systems – Con- formal densities and Hausdorff dimension. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Hyperbolic groups and nonpositively curved groups. msc | Group theory and generalizations – Structure and classification of infinite or finite groups – Groups acting on trees. msc | Dynamical systems and ergodic theory – Ergodic theory – Relations with number theory and harmonic analysis. msc | Topological groups, Lie groups – Lie groups – Infinite-dimensional Lie groups and their Lie algebras: general properties. msc | Group theory and generalizations – Semigroups – Semigroups of transformations, etc.. msc Classification: LCC QA685 .D238 2017 | DDC 516.3/62–dc23 LC record available at https://lccn. loc.gov/2016034629
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Contents
List of Figures xi
Prologue xiii
Chapter 1. Introduction and Overview xvii 1.1. Preliminaries xviii 1.1.1. Algebraic hyperbolic spaces xviii 1.1.2. Gromov hyperbolic metric spaces xviii 1.1.3. Discreteness xxi 1.1.4. The classification of semigroups xxii 1.1.5. Limit sets xxiii 1.2. The Bishop–Jones theorem and its generalization xxiv 1.2.1. The modified Poincar´e exponent xxvii 1.3. Examples xxviii 1.3.1. Schottky products xxviii 1.3.2. Parabolic groups xxix 1.3.3. Geometrically finite and convex-cobounded groups xxix 1.3.4. Counterexamples xxx 1.3.5. R-trees and their isometry groups xxxi 1.4. Patterson–Sullivan theory xxxi 1.4.1. Quasiconformal measures of geometrically finite groups xxxiv 1.5. Appendices xxxv
Part 1. Preliminaries 1
Chapter 2. Algebraic hyperbolic spaces 3 2.1. The definition 3 2.2. The hyperboloid model 4 2.3. Isometries of algebraic hyperbolic spaces 7 2.4. Totally geodesic subsets of algebraic hyperbolic spaces 12 2.5. Other models of hyperbolic geometry 15 2.5.1. The (Klein) ball model 16 2.5.2. The half-space model 16 2.5.3. Transitivity of the action of Isom(H)on∂H 18
Chapter 3. R-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces 19 3.1. Graphs and R-trees 19 3.2. CAT(-1) spaces 22 3.2.1. Examples of CAT(-1) spaces 23 3.3. Gromov hyperbolic metric spaces 24
vii viii CONTENTS
3.3.1. Examples of Gromov hyperbolic metric spaces 26 3.4. The boundary of a hyperbolic metric space 27 3.4.1. Extending the Gromov product to the boundary 29 3.4.2. A topology on bord X 32 3.5. The Gromov product in algebraic hyperbolic spaces 35 3.5.1. The Gromov boundary of an algebraic hyperbolic space 39 3.6. Metrics and metametrics on bord X 40 3.6.1. General theory of metametrics 40 3.6.2. The visual metametric based at a point w ∈ X 42 3.6.3. The extended visual metric on bord X 43 3.6.4. The visual metametric based at a point ξ ∈ ∂X 45
Chapter 4. More about the geometry of hyperbolic metric spaces 49 4.1. Gromov triples 49 4.2. Derivatives 50 4.2.1. Derivatives of metametrics 50 4.2.2. Derivatives of maps 51 4.2.3. The dynamical derivative 53 4.3. The Rips condition 54 4.4. Geodesics in CAT(-1) spaces 55 4.5. The geometry of shadows 61 4.5.1. Shadows in regularly geodesic hyperbolic metric spaces 61 4.5.2. Shadows in hyperbolic metric spaces 61 4.6. Generalized polar coordinates 66
Chapter 5. Discreteness 69 5.1. Topologies on Isom(X)69 5.2. Discrete groups of isometries 72 5.2.1. Topological discreteness 74 5.2.2. Equivalence in finite dimensions 76 5.2.3. Proper discontinuity 76 5.2.4. Behavior with respect to restrictions 78 5.2.5. Countability of discrete groups 78
Chapter 6. Classification of isometries and semigroups 79 6.1. Classification of isometries 79 6.1.1. More on loxodromic isometries 81 6.1.2. The story for real hyperbolic spaces 82 6.2. Classification of semigroups 82 6.2.1. Elliptic semigroups 83 6.2.2. Parabolic semigroups 83 6.2.3. Loxodromic semigroups 84 6.3. Proof of the Classification Theorem 85 6.4. Discreteness and focal groups 87
Chapter 7. Limit sets 91 7.1. Modes of convergence to the boundary 91 7.2. Limit sets 93 7.3. Cardinality of the limit set 95 7.4. Minimality of the limit set 96 CONTENTS ix
7.5. Convex hulls 99 7.6. Semigroups which act irreducibly on algebraic hyperbolic spaces 102 7.7. Semigroups of compact type 103
Part 2. The Bishop–Jones theorem 107
Chapter 8. The modified Poincar´e exponent 109 8.1. The Poincar´e exponent of a semigroup 109 8.2. The modified Poincar´e exponent of a semigroup 110
Chapter 9. Generalization of the Bishop–Jones theorem 115 9.1. Partition structures 116 9.2. A partition structure on ∂X 120 9.3. Sufficient conditions for Poincar´e regularity 127
Part 3. Examples 131
Chapter 10. Schottky products 133 10.1. Free products 133 10.2. Schottky products 134 10.3. Strongly separated Schottky products 135 10.4. A partition-structure–like structure 142 10.5. Existence of Schottky products 146
Chapter 11. Parabolic groups 149 11.1. Examples of parabolic groups acting on E∞ 149 11.1.1. The Haagerup property and the absence of a Margulis lemma 150 11.1.2. Edelstein examples 151 11.2. The Poincar´e exponent of a finitely generated parabolic group 155 11.2.1. Nilpotent and virtually nilpotent groups 156 11.2.2. A universal lower bound on the Poincar´e exponent 157 11.2.3. Examples with explicit Poincar´e exponents 158
Chapter 12. Geometrically finite and convex-cobounded groups 165 12.1. Some geometric shapes 165 12.1.1. Horoballs 165 12.1.2. Dirichlet domains 167 12.2. Cobounded and convex-cobounded groups 168 12.2.1. Characterizations of convex-coboundedness 170 12.2.2. Consequences of convex-coboundedness 172 12.3. Bounded parabolic points 172 12.4. Geometrically finite groups 176 12.4.1. Characterizations of geometrical finiteness 177 12.4.2. Consequences of geometrical finiteness 182 12.4.3. Examples of geometrically finite groups 186
Chapter 13. Counterexamples 189 13.1. Embedding R-trees into real hyperbolic spaces 189 13.2. Strongly discrete groups with infinite Poincar´e exponent 193 13.3. Moderately discrete groups which are not strongly discrete 193 xCONTENTS
13.4. Poincar´e irregular groups 194 13.5. Miscellaneous counterexamples 198
Chapter 14. R-trees and their isometry groups 199 14.1. Construction of R-trees by the cone method 199 14.2. Graphs with contractible cycles 202 14.3. The nearest-neighbor projection onto a convex set 204 14.4. Constructing R-trees by the stapling method 205 14.5. Examples of R-trees constructed using the stapling method 209
Part 4. Patterson–Sullivan theory 217 Chapter 15. Conformal and quasiconformal measures 219 15.1. The definition 219 15.2. Conformal measures 220 15.3. Ergodic decomposition 220 15.4. Quasiconformal measures 222 15.4.1. Pointmass quasiconformal measures 223 15.4.2. Non-pointmass quasiconformal measures 224 Chapter 16. Patterson–Sullivan theorem for groups of divergence type 229 16.1. Samuel–Smirnov compactifications 229 16.2. Extending the geometric functions to X 230 16.3. Quasiconformal measures on X 232 16.4. The main argument 234 16.5. End of the argument 237 16.6. Necessity of the generalized divergence type assumption 238 16.7. Orbital counting functions of nonelementary groups 239 Chapter 17. Quasiconformal measures of geometrically finite groups 241 17.1. Sufficient conditions for divergence type 241 17.2. The global measure formula 244 17.3. Proof of the global measure formula 247 17.4. Groups for which μ is doubling 253 17.5. Exact dimensionality of μ 259 17.5.1. Diophantine approximation on Λ 261 17.5.2. Examples and non-examples of exact dimensional measures 264 Appendix A. Open problems 267 Appendix B. Index of defined terms 269 Bibliography 275 List of Figures
3.1.1 A geodesic triangle in an R-tree 23 3.3.1 A quadruple of points in an R-tree 25 3.3.2 Expressing distance via Gromov products in an R-tree 26 3.4.1 A Gromov sequence in an R-tree 28 3.5.1 Relating angle and the Gromov product 35 3.5.2 B is strongly Gromov hyperbolic 38 3.5.3 A formula for the Busemann function in the half-space model 40 3.6.1 The Hamenst¨adt distance 47
4.2.1 The derivative of g at ∞ 52 4.3.1 The Rips condition 55
4.4.1 The triangle Δ(x, y1,y2)58 4.5.1 Shadows in regularly geodesic hyperbolic metric spaces 62 4.5.2 The Intersecting Shadows Lemma 63 4.5.3 The Big Shadows Lemma 64 4.5.4 The Diameter of Shadows Lemma 65 4.6.1 Polar coordinates in the half-space model 66
6.4.1 High altitude implies small displacement in the half-space model 89
7.1.1 Conical convergence to the boundary 91 7.1.2 Converging horospherically but not radially to the boundary 93
9.2.1 The construction of children 121
9.2.2 The sets Cn,forn ∈ Z 122
10.3.1 The strong separation lemma for Schottky products 137
12.1.1 Visualizing horoballs in the ball and half-space models 166 12.1.2 Diameter decay of a ball complement inside a horoball 166
12.1.3 The Cayley graph of Γ = F2(Z)=γ1,γ2 168 12.2.1 Proving that convex-cobounded groups are of compact type 171 12.3.1 The geometry of bounded parabolic points 175 12.4.1 Proving that geometrically finite groups are of compact type 179
xi xii LIST OF FIGURES
12.4.2 Local finiteness of the horoball collection 180 12.4.3 Orbit maps of geometrically finite groups are QI embeddings 184
13.4.1 Geometry of automorphisms of a simplicial tree 195
14.2.1 Triangles in graphs with contractible cycles 204 14.2.2 Triangles in graphs with contractible cycles: general case 204 14.4.1 The consistency condition for stapling metric spaces 207
14.5.1 The Cayley graph of F2(Z) as a pure Schottky product 212 14.5.2 An example of a geometric product 214 14.5.3 Another example of a geometric product 215
17.2.1 Cusp excursion and ball measure functions 245 17.3.1 Estimating measures of balls via information “at infinity” 249 17.3.2 Estimating measures of balls via “local” information 251 Prologue
...Cela suffit pour faire comprendre que dans les cinq m´emoires des Acta mathematica que j’ai consacr´es `al’´etude des transcen- dantes fuchsiennes et klein´eennes, je n’ai fait qu’effleurer un su- jet tr`es vaste, qui fournira sans doute aux g´eom`etres l’occasion de nombreuses et importantes d´ecouvertes.1 – H. Poincar´e, Acta Mathematica, 5, 1884, p. 278. The theory of discrete subgroups of real hyperbolic space has a long history. It was inaugurated by Poincar´e, who developed the two-dimensional (Fuchsian) and three-dimensional (Kleinian) cases of this theory in a series of articles published between 1881 and 1884 that included numerous notes submitted to the C. R. Acad. Sci. Paris, a paper at Klein’s request in Math. Annalen, and five memoirs com- missioned by Mittag-Leffler for his then freshly-minted Acta Mathematica. One must also mention the complementary work of the German school that came be- fore Poincar´e and continued well after he had moved on to other areas, viz. that of Klein, Schottky, Schwarz, and Fricke. See [80, Chapter 3] for a brief exposi- tion of this fascinating history, and [79, 63] for more in-depth presentations of the mathematics involved. We note that in finite dimensions, the theory of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic d-space Hd for d ≥ 4, is markedly different from that in H3 and H2. For example, the Teichm¨uller the- ory used by the Ahlfors–Bers school (viz. Marden, Maskit, Jørgensen, Sullivan, Thurston, etc.) to study three-dimensional Kleinian groups has no generalization to higher dimensions. Moreover, the recent resolution of the Ahlfors measure con- jecture [3, 43] has more to do with three-dimensional topology than with analysis and dynamics. Indeed, the conjecture remains open in higher dimensions [106,p. 526, last paragraph]. Throughout the twentieth century, there are several instances of theorems proven for three-dimensional Kleinian groups whose proofs extended easily to n dimensions (e.g. [21, 133]), but it seems that the theory of higher- dimensional Kleinian groups was not really considered a subject in its own right until around the 1990s. For more information on the theory of higher-dimensional Kleinian groups, see the survey article [106], which describes the state of the art up to the last decade, emphasizing connections with homological algebra.
1This is enough to make it apparent that in these five memoirs in Acta Mathematica which I have dedicated to the study of Fuschian and Kleinian transcendants, I have only skimmed the surface of a very broad subject, which will no doubt provide geometers with the opportunity for many important discoveries.
xiii xiv PROLOGUE
But why stop at finite n? Dennis Sullivan, in his IHES´ Seminar on Conformal and Hyperbolic Geometry [164] that ran during the late 1970s and early ’80s, indi- cated a possibility of developing the theory of discrete groups acting by hyperbolic isometries on the open unit ball of a separable infinite-dimensional Hilbert space.2 Later in the early ’90s, Misha Gromov observed the paucity of results regarding such actions in his seminal lectures Asymptotic Invariants of Infinite Groups [86] where he encouraged their investigation in memorable terms: “The spaces like this [infinite-dimensional symmetric spaces] . . . look as cute and sexy to me as their finite dimensional siblings but they have been for years shamefully neglected by geometers and algebraists alike”. Gromov’s lament had not fallen to deaf ears, and the geometry and represen- tation theory of infinite-dimensional hyperbolic space H∞ and its isometry group have been studied in the last decade by a handful of mathematicians, see e.g. [40, 65, 132]. However, infinite-dimensional hyperbolic geometry has come into prominence most spectacularly through the recent resolution of a long-standing conjecture in algebraic geometry due to Enriques from the late nineteenth cen- tury. Cantat and Lamy [47] proved that the Cremona group (i.e. the group of birational transformations of the complex projective plane) has uncountably many non-isomorphic normal subgroups, thus disproving Enriques’ conjecture. Key to their enterprise is the fact, due to Manin [125], that the Cremona group admits a faithful isometric action on a non-separable infinite-dimensional hyperbolic space, now known as the Picard–Manin space. Our project was motivated by a desire to answer Gromov’s plea by exposing a coherent general theory of groups acting isometrically on the infinite-dimensional hyperbolic space H∞. In the process we came to realize that a more natural do- main for our inquiries was the much larger setting of semigroups acting on Gro- mov hyperbolic metric spaces – that way we could simultaneously answer our own questions about H∞ and construct a theoretical framework for those who are in- terested in more exotic spaces such as the curve graph, arc graph, and arc complex [95, 126, 96] and the free splitting and free factor complexes [89, 27, 104, 96]. These examples are particularly interesting as they extend the well-known dic- tionary [26, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging between mapping class groups and Cre- mona groups, see [30, 66]. We speculate that developing the Patterson–Sullivan theory in these three areas would be fruitful and may lead to new connections and analogies that have not surfaced till now. In a similar spirit, we believe there is a longer story for which this mono- graph lays the foundations. In general, infinite-dimensional space is a wellspring of outlandish examples and the wide range of new phenomena we have started to uncover has no analogue in finite dimensions. The geometry and analysis of such groups should pique the interests of specialists in probability, geometric group theory, and metric geometry. More speculatively, our work should interact with the ongoing and still nascent study of geometry, topology, and dynamics in a vari- ety of infinite-dimensional spaces and groups, especially in scenarios with sufficient
2This was the earliest instance of such a proposal that we could find in the literature, although (as pointed out to us by P. de la Harpe) infinite-dimensional hyperbolic spaces without groups acting on them had been discussed earlier [130, §27], [131, 60]. It would be of interest to know whether such an idea may have been discussed prior to that. PROLOGUE xv negative curvature. Here are three concrete settings that would be interesting to consider: the universal Teichm¨uller space, the group of volume-preserving diffeo- morphisms of R3 or a 3-torus, and the space of K¨ahler metrics/potentials on a closed complex manifold in a fixed cohomology class equipped with the Mabuchi– Semmes–Donaldson metric. We have been developing a few such themes. The study of thermodynamics (equilibrium states and Gibbs measures) on the boundaries of Gromov hyperbolic spaces will be investigated in future work [57]. We speculate that the study of stochastic processes (random walks and Brownian motion) in such settings would be fruitful. Furthermore, it would be of interest to develop the theory of discrete isometric actions and limit sets in infinite-dimensional spaces of higher rank.
Acknowledgements. This monograph is dedicated to our colleague Bernd O. Stratmann, who passed away on the 8th of August, 2015. Various discussions with Bernd provided inspiration for this project and we remain grateful for his friendship. The authors thank D. P. Sullivan, D. Mumford, B. Farb, P. Pansu, F. Ledrappier, A. Wilkinson, K. Biswas, E. Breuillard, A. Karlsson, I. Assani, M. Lapidus, R. Guo, Z. Huang, I. Gekhtman, G. Tiozzo, P. Py, M. Davis, M. Roychowdury, M. Hochman, J. Tao, P. de la Harpe, T. Barthelme, J. P. Conze, and Y. Guivarc’h for their interest and encouragement, as well as for invitations to speak about our work at various venues. We are grateful to S. J. Patterson, J. Elstrodt, and E.´ Ghys for enlightening historical discussions on various themes relating to the history of Fuchsian and Kleinian groups and their study through the twentieth century, and to D. P. Sullivan and D. Mumford for suggesting work on diffeomorphism groups and the universal Teichm¨uller space. We thank X. Xie for pointing us to a solution to one of the problems in our Appendix A. The research of the first-named author was supported in part by 2014-2015 and 2016-2017 Faculty Research Grants from the University of Wisconsin–La Crosse. The research of the second-named author was supported in part by the EPSRC Programme Grant EP/J018260/1. The research of the third-named author was supported in part by the NSF grant DMS-1361677.
CHAPTER 1
Introduction and Overview
The purpose of this monograph is to present the theory of groups and semi- groups acting isometrically on Gromov hyperbolic metric spaces in full detail as we understand it, with special emphasis on the case of infinite-dimensional alge- ∞ braic hyperbolic spaces X = HF ,whereF denotes a division algebra. We have not skipped over the parts which some would call “trivial” extensions of the finite- dimensional/proper theory, for two main reasons: first, intuition has turned out to be wrong often enough regarding these matters that we feel it is worth writing ev- erything down explicitly; second, we feel it is better methodologically to present the entire theory from scratch, in order to provide a basic reference for the theory, since no such reference exists currently (the closest, [39], has a fairly different emphasis). Thus Part 1 of this monograph should be treated as mostly expository, while Parts 2-4 contain a range of new material. For experts who want a geodesic path to significant theorems, we list here five such results that we prove in this monograph: Theorems 1.2.1 and 1.4.4 provide generalizations of the Bishop–Jones theorem [28, Theorem 1] and the Global Measure Formula [160, Theorem 2], respectively, to Gromov hyperbolic metric spaces. Theorem 1.4.1 guarantees the existence of a δ-quasiconformal measure for groups of divergence type, even if the space they are acting on is not proper. Theorem 1.4.5 provides a sufficient condition for the exact dimensionality of the Patterson-Sullivan measure of a geometrically finite group, and Theorem 1.4.6 relates the exact dimensionality to Diophantine properties of the measure. However, the reader should be aware that a sharp focus on just these results, without care for their motivation or the larger context in which they are sit- uated, will necessarily preclude access to the interesting and uncharted landscapes that our work has begun to uncover. The remainder of this chapter provides an overview of these landscapes.
Convention 1. The symbols , ,and will denote coarse asymptotics; a subscript of + indicates that the asymptotic is additive, and a subscript of × indicates that it is multiplicative. For example, A ×,K B means that there exists a constant C>0 (the implied constant), depending only on K, such that A ≤ CB. Moreover, A +,× B means that there exist constants C1,C2 > 0sothatA ≤ C1B + C2. In general, dependence of the implied constant(s) on universal objects suchasthemetricspaceX, the group G, and the distinguished point o ∈ X (cf. Notation 1.1.5) will be omitted from the notation.
Convention 2. The notation xn −→ x means that xn → x as n →∞, while n the notation xn −−→ x means that n,+
x + lim sup xn + lim inf xn, n→∞ n→∞
xvii xviii 1. INTRODUCTION AND OVERVIEW and similarly for xn −−→ x. n,× Convention 3. The symbol is used to indicate the end of a nested proof. Convention . 4 We use the Iverson bracket notation: 1 statement true [statement] = 0 statement false Convention 5. Given a distinguished point o ∈ X,wewrite x = d(o, x)and g = g(o) .
1.1. Preliminaries 1.1.1. Algebraic hyperbolic spaces. Although we are mostly interested in ∞ this monograph in the real infinite-dimensional hyperbolic space HR , the complex ∞ ∞ and quaternionic hyperbolic spaces HC and HQ are also interesting. In finite dimensions, these spaces constitute (modulo the Cayley hyperbolic plane1)the rank one symmetric spaces of noncompact type. In the infinite-dimensional case we retain this terminology by analogy; cf. Remark 2.2.6. For brevity we will refer to a rank one symmetric space of noncompact type as an algebraic hyperbolic space. There are several equivalent ways to define algebraic hyperbolic spaces; these are known as “models” of hyperbolic geometry. We consider here the hyperboloid model, ball model (Klein’s, not Poincar´e’s), and upper half-space model (which only applies to algebraic hyperbolic spaces defined over the reals, which we will call α α α real hyperbolic spaces),whichwedenotebyHF , BF ,andE , respectively. Here F denotes the base field (either R, C,orQ), and α denotes a cardinal number. We omit the base field when it is R, and denote the exponent by ∞ when it is #(N), ∞ #(N) so that H = HR is the unique separable infinite-dimensional real hyperbolic space. The main theorem of Chapter 2 is Theorem 2.3.3, which states that any isom- etry of an algebraic hyperbolic space must be an “algebraic” isometry. The finite- dimensional case is given as an exercise in Bridson–Haefliger [39, Exercise II.10.21]. We also describe the relation between totally geodesic subsets of algebraic hyper- bolic spaces and fixed point sets of isometries (Theorem 2.4.7), a relation which will be used throughout the paper. Remark 1.1.1. Key to the study of finite-dimensional algebraic hyperbolic spaces is the theory of quasiconformal mappings (e.g., as in Mostow and Pansu’s rigidity theorems [133, 141]). Unfortunately, it appears to be quite difficult to generalize this theory to infinite dimensions. For example, it is an open question [92, p.1335] whether every quasiconformal homeomorphism of Hilbert space is also quasisymmetric. 1.1.2. Gromov hyperbolic metric spaces. Historically, the first motiva- tion for the theory of negatively curved metric spaces came from differential ge- ometry and the study of negatively curved Riemannian manifolds. The idea was to describe the most important consequences of negative curvature in terms of the metric structure of the manifold. This approach was pioneered by Aleksandrov
1 2 We omit all discussion of the Cayley hyperbolic plane HO, as the algebra involved is too exotic for our taste; cf. Remark 2.1.1. 1.1. PRELIMINARIES xix
[6], who discovered for each κ ∈ R an inequality regarding triangles in a metric space with the property that a Riemannian manifold satisfies this inequality if and only if its sectional curvature is bounded above by κ, and popularized by Gromov, who called Aleksandrov’s inequality the “CAT(κ) inequality” as an abbreviation for “comparison inequality of Alexandrov–Toponogov” [85, p.106].2 A metric space is called CAT(κ) if the distance between any two points on a geodesic triangle is smaller than the corresponding distance on the “comparison triangle” in a model space of constant curvature κ; see Definition 3.2.1. The second motivation came from geometric group theory, in particular the study of groups acting on manifolds of negative curvature. For example, Dehn proved that the word problem is solvable for finitely generated Fuchsian groups [64], and this was generalized by Cannon to groups acting cocompactly on manifolds of negative curvature [44]. Gromov attempted to give a geometric characterization of these groups in terms of their Cayley graphs; he tried many definitions (cf. [83, §6.4], [84, §4]) before converging to what is now known as Gromov hyperbolicity in 1987 [85, 1.1, p.89], a notion which has influenced much research. A metric space is said to be Gromov hyperbolic if it satisfies a certain inequality that we call Gromov’s inequality; see Definition 3.3.2. A finitely generated group is then said to be word-hyperbolic if its Cayley graph is Gromov hyperbolic. The big advantage of Gromov hyperbolicity is its generality. We give some idea of its scope by providing the following nested list of metric spaces which have been proven to be Gromov hyperbolic: • CAT(-1) spaces (Definition 3.2.1) – Riemannian manifolds (both finite- and infinite-dimensional) with sectional curvature ≤−1 ∗ Algebraic hyperbolic spaces (Definition 2.2.5) · Picard–Manin spaces of projective surfaces defined over algebraically closed fields [125], cf. [46, §3.1] – R-trees (Definition 3.1.10) ∗ Simplicial trees · Unweighted simplicial trees • Cayley metrics (Example 3.1.2) on word-hyperbolic groups • Green metrics on word-hyperbolic groups [29, Corollary 1.2] • Quasihyperbolic metrics of uniform domains in Banach spaces [173,The- orem 2.12] • Arc graphs and curve graphs [95] and arc complexes [126, 96] of finitely punctured oriented surfaces • Free splitting complexes [89, 96] and free factor complexes [27, 104, 96] Remark 1.1.2. Many of the above examples admit natural isometric group actions: • The Cremona group acts isometrically on the Picard–Manin space [125], cf. [46, Theorem 3.3]. • The mapping class group of a finitely punctured oriented surface acts isometrically on its arc graph, curve graph, and arc complex.
2It appears that Bridson and Haefliger may be responsible for promulgating the idea that the C in CAT refers to E. Cartan [39, p.159]. We were unable to find such an indication in [85], although Cartan is referenced in connection with some theorems regarding CAT(κ) spaces (as are Riemann and Hadamard). xx 1. INTRODUCTION AND OVERVIEW
• The outer automorphism group Out(FN ) of the free group on N generators acts isometrically on the free splitting complex FS(FN ) and the free factor complex FF(FN ). Remark 1.1.3. Most of the above examples are examples of non-proper hyper- bolic metric spaces. Recall that a metric space is said to be proper if its distance function x → x = d(o, x) is proper, or equivalently if closed balls are compact. Though much of the existing literature on CAT(-1) and hyperbolic metric spaces assumes that the spaces in question are proper, it is often not obvious whether this assumption is really essential. However, since results about proper metric spaces do not apply to infinite-dimensional algebraic hyperbolic spaces, we avoid the as- sumption of properness. Remark 1.1.4. One of the above examples, namely, Green metrics on word- hyperbolic groups, is a natural class of non-geodesic hyperbolic metric spaces.3 However, Bonk and Schramm proved that all non-geodesic hyperbolic metric spaces can be isometrically embedded into geodesic hyperbolic metric spaces [31,Theorem 4.1], and the equivariance of their construction was proven by Blach`ere, Ha¨ıssinsky, and Mathieu [29, Corollary A.10]. Thus, one could view the assumption of geodesic- ity to be harmless, since most theorems regarding geodesic hyperbolic metric spaces can be pulled back to non-geodesic hyperbolic metric spaces. However, for the most part we also avoid the assumption of geodesicity, mostly for methodological reasons rather than because we are considering any particular non-geodesic hyperbolic met- ric space. Specifically, we felt that Gromov’s definition of hyperbolicity in metric spaces is a “deep” definition whose consequences should be explored independently of such considerations as geodesicity. We do make the assumption of geodesic- ity in Chapter 12, where it seems necessary in order to prove the main theorems. (The assumption of geodesicity in Chapter 12 can for the most part be replaced by the weaker assumption of almost geodesicity [31, p.271], but we felt that such a presentation would be more technical and less intuitive.) We now introduce a list of standing assumptions and notations. They apply to all chapters except for Chapters 2, 3, and 5 (see also §4.1). Notation 1.1.5. Throughout the introduction, • X is a Gromov hyperbolic metric space (cf. Definition 3.3.2), • d denotes the distance function of X, • ∂X denotes the Gromov boundary of X,andbordX denotes the bordifi- cation bord X = X ∪ ∂X (cf. Definition 3.4.2), • D denotes a visual metric on ∂X with respect to a parameter b>1anda distinguished point o ∈ X (cf. Proposition 3.6.8). By definition, a visual metric satisfies the asymptotic
−ξ|ηo (1.1.1) Db,o(ξ,η) × b , where ·|· denotes the Gromov product (cf. (3.3.2)). • Isom(X) denotes the isometry group of X.Also,G ≤ Isom(X) will mean that G is a subgroup of Isom(X), while G Isom(X) will mean that G is a subsemigroup of Isom(X).
3Quasihyperbolic metrics on uniform domains in Banach spaces can also fail to be geodesic, but they are almost geodesic which is almost as good. See e.g. [172] for a study of almost geodesic hyperbolic metric spaces. 1.1. PRELIMINARIES xxi
A prime example to have in mind is the special case where X is an infinite- dimensional algebraic hyperbolic space, in which case the Gromov boundary ∂X can be identified with the natural boundary of X (Proposition 3.5.3), and we can set b = e and get equality in (1.1.1) (Observation 3.6.7). Another important example of a hyperbolic metric space that we will keep in our minds is the case of R-trees alluded to above. R-trees are a generalization of simplicial trees, which in turn are a generalization of unweighted simplicial trees, also known as “Z-trees” or just “trees”. R-trees are worth studying in the context of hyperbolic metric spaces for two reasons: first of all, they are “prototype spaces” in the sense that any finite set in a hyperbolic metric space can be roughly iso- metrically embedded into an R-tree, with a roughness constant depending only on the cardinality of the set [77, pp.33-38]; second of all, R-trees can be equivariantly embedded into infinite-dimensional real hyperbolic space H∞ (Theorem 13.1.1), meaning that any example of a group acting on an R-tree can be used to construct an example of the same group acting on H∞. R-trees are also much simpler to understand than general hyperbolic metric spaces: for any finite set of points, one can draw out a list of all possible diagrams, and then the set of distances must be determined from one of these diagrams (cf. e.g., Figure 3.3.1). Besides introducing R-trees, CAT(-1) spaces, and hyperbolic metric spaces, the following things are done in Chapter 3: construction of the Gromov boundary ∂X and analysis of its basic topological properties (Section 3.4), proof that the Gromov boundary of an algebraic hyperbolic space is equal to its natural boundary (Proposition 3.5.3), and the construction of various metrics and metametrics on the boundary of X (Section 3.6). None of this is new, although the idea of a metametric (due to V¨ais¨al¨a[172, §4])isnotverywellknown. In Chapter 4, we go more into detail regarding the geometry of hyperbolic metric spaces. We prove the geometric mean value theorem for hyperbolic metric spaces (Section 4.2), the existence of geodesic rays connecting two points in the boundary of a CAT(-1) space (Proposition 4.4.4), and various geometrical theorems regarding the sets
Shadz(x, σ):={ξ ∈ ∂X : x|ξz ≤ σ}, which we call “shadows” due to their similarity to the famous shadows of Sullivan [161, Fig. 2] on the boundary of Hd (Section 4.5). We remark that most proofs of the existence of geodesics between points on the boundary of complete CAT(-1) spaces, e.g. [39, Proposition II.9.32], assume properness and make use of it in a crucial way, whereas we make no such assumption in Proposition 4.4.4. Finally, in Section 4.6 we introduce “generalized polar coordinates” in a hyperbolic metric space. These polar coordinates tell us that the action of a loxodromic isometry (see Definition 6.1.2) on a hyperbolic metric space is roughly the same as the map x → λx in the upper half-plane E2.
1.1.3. Discreteness. The first step towards extending the theory of Kleinian groups to infinite dimensions (or more generally to hyperbolic metric spaces) is to define the appropriate class of groups to consider. This is less trivial than might be expected. Recalling that a d-dimensional Kleinian group is defined to be a discrete subgroup of Isom(Hd), we would want to define an infinite-dimensional Kleinian group to be a discrete subgroup of Isom(H∞). But what does it mean for a subgroup of Isom(H∞) to be discrete? In finite dimensions, the most natural definition is xxii 1. INTRODUCTION AND OVERVIEW to call a subgroup discrete if it is discrete relative to the natural topology on Isom(Hd); this definition works well since Isom(Hd) is a Lie group. But in infinite dimensions and especially in more exotic spaces, many applications require stronger hypotheses (e.g., Theorem 1.2.1, Chapter 12). In Chapter 5, we discuss several potential definitions of discreteness, which are inequivalent in general but agree in the case of finite-dimensional space X = Hd (Proposition 5.2.10): Definitions 5.2.1 and 5.2.6. Fix G ≤ Isom(X). • G is called strongly discrete (SD) if for every bounded set B ⊆ X,wehave #{g ∈ G : g(B) ∩ B = } < ∞. • G is called moderately discrete (MD) if for every x ∈ X,thereexistsan open set U containing x such that #{g ∈ G : g(U) ∩ U = } < ∞. • G is called weakly discrete (WD) if for every x ∈ X, there exists an open set U containing x such that g(U) ∩ U = ⇒ g(x)=x. • G is called COT-discrete (COTD) if it is discrete as a subset of Isom(X) when Isom(X) is given the compact-open topology (COT). • If X is an algebraic hyperbolic space, then G is called UOT-discrete (UOTD) if it is discrete as a subset of Isom(X)whenIsom(X)isgiven the uniform operator topology (UOT; cf. Section 5.1). As our naming suggests, the condition of strong discreteness is stronger than the condition of moderate discreteness, which is in turn stronger than the condition of weak discreteness (Proposition 5.2.4). Moreover, any moderately discrete group is COT-discrete, and any weakly discrete subgroup of Isom(H∞) is COT-discrete (Proposition 5.2.7). These relations and more are summarized in Table 1 on p. 77. Out of all these definitions, strong discreteness should perhaps be thought of as the best generalization of discreteness to infinite dimensions. Thus, we propose that the phrase “infinite-dimensional Kleinian group” should mean “strongly discrete subgroup of Isom(H∞)”. However, in this monograph we will be interested in the consequences of all the different notions of discreteness, as well as the interactions between them. Remark 1.1.6. Strongly discrete groups are known in the literature as met- rically proper, and moderately discrete groups are known as wandering. However, we prefer our terminology since it more clearly shows the relationship between the different notions of discreteness.
1.1.4. The classification of semigroups. After clarifying the different types of discreteness which can occur in infinite dimensions, we turn to the question of classification. This question makes sense both for individual isometries and for en- tire semigroups.4 Historically, the study of classification began in the 1870s when
4In Chapters 6-10, we work in the setting of semigroups rather than groups. Like dropping the assumption of geodesicity (cf. Remark 1.1.4), this is done partly in order to broaden our class of examples and partly for methodological reasons – we want to show exactly where the assumption of being closed under inverses is being used. It should be also noted that semigroups sometimes show up naturally when one is studying groups; cf. Proposition 10.5.4(B). 1.1. PRELIMINARIES xxiii
Klein proved a theorem classifying isometries of H2 and attached the words “ellip- tic”, “parabolic”, and “hyperbolic” to these classifications. Elliptic isometries are those which have at least one fixed point in the interior, while parabolic isometries have exactly one fixed point, which is a neutral fixed point on the boundary, and hyperbolic isometries have two fixed points on the boundary, one of which is at- tracting and one of which is repelling. Later, the word “loxodromic” was used to refer to isometries in H3 which have two fixed points on the boundary but which are geometrically “screw motions” rather than simple translations. In what follows we use the word “loxodromic” to refer to all isometries of Hn (or more generally a hyperbolic metric space) with two fixed points on the boundary – this is analogous to calling a circle an ellipse. Our real reason for using the word “loxodromic” in this instance, rather than “hyperbolic”, is to avoid confusion with the many other meanings of the word “hyperbolic” that have entered usage in various scenarios. To extend this classification from individual isometries to groups, we call a group “elliptic” if its orbits are bounded, “parabolic” if it has a unique neutral global fixed point on the boundary, and “loxodromic” if it contains at least one loxodromic isometry. The main theorem of Chapter 6 (viz. Theorem 6.2.3) is that every subsemigroup of Isom(X) is either elliptic, parabolic, or loxodromic. Classification of groups has appeared in the literature in various contexts, from Eberlein and O’Neill’s results regarding visiblility manifolds [69], through Gro- mov’s remarks about groups acting on strictly convex spaces [83, §3.5] and word- hyperbolic groups [85, §3.1], to the more general results of Hamann [88,Theorem 2.7], Osin [140, §3], and Caprace, de Cornulier, Monod, and Tessera [48, §3.A] regarding geodesic hyperbolic metric spaces.5 Many of these theorems have similar statements to ours ([88]and[48] seem to be the closest), but we have not kept track of this carefully, since our proof appears to be sufficiently different to warrant independent interest anyway. After proving Theorem 6.2.3, we discuss further aspects of the classification of groups, such as the further classification of loxodromic groups given in §6.2.3: a loxodromic group is called “lineal”, “focal”, or “of general type” according to whether it has two, one, or zero global fixed points, respectively. (This terminology was introduced in [48].) The “focal” case is especially interesting, as it represents a class of nonelementary groups which have global fixed points.6 We show that certain classes of discrete groups cannot be focal (Proposition 6.4.1), which explains why such groups do not appear in the theory of Kleinian groups. On the other hand, we show that in infinite dimensions, focal groups can have interesting limit sets even though they satisfy only a weak form of discreteness; cf. Remark 13.4.3.
1.1.5. Limit sets. An important invariant of a Kleinian group G is its limit set Λ=ΛG, the set of all accumulation points of the orbit of any point in the interior. By putting an appropriate topology on the bordification of our hyperbolic metric space X (§3.4.2), we can generalize this definition to an arbitrary subsemi-
5We remark that the results of [48, §3.A] can be generalized to non-geodesic hyperbolic metric spaces by using the Bonk–Schramm embedding theorem [31, Theorem 4.1] (see also [29, Corollary A.10]). 6Some sources (e.g. [148, §5.5]) define nonelementarity in a way such that global fixed points are automatically ruled out, but this is not true of our definition (Definition 7.3.2). xxiv 1. INTRODUCTION AND OVERVIEW group of Isom(X). Many results generalize relatively straightforwardly7 to this new context, such as the minimality of the limit set (Proposition 7.4.1) and the connec- tion between classification and the cardinality of the limit set (Proposition 7.3.1). In particular, we call a semigroup elementary if its limit set is finite. In general, the convex hull of the limit set may need to be replaced by a quasiconvex hull (cf. Definition 7.5.1), since in certain cases the convex hull does not accurately reflect the geometry of the group. Indeed, Ancona [9, Corollary C] and Borbely [32, Theorem 1] independently constructed examples of CAT(-1) three-manifolds X for which there exists a point ξ ∈ ∂X such that the convex hull of any neighborhood of ξ is equal to bord X. Although in a non-proper setting the limit set may no longer be compact, compactness of the limit set is a reasonable geometric condition that is satisfied for many examples of subgroups of Isom(H∞) (e.g. Examples 13.2.2, 13.4.2). We call this condition compact type (Definition 7.7.1).
1.2. The Bishop–Jones theorem and its generalization The term Poincar´eseriesclassically referred to a variety of averaging pro- cedures, initiated by Poincar´e in his aforementioned Acta memoirs, with a view towards uniformization of Riemann surfaces via the construction of automorphic forms. Given a Fuchsian group Γ and a rational function H : C → C with no poles on ∂B2, Poincar´e proved that for every m ≥ 2theseries H(γ(z))(γ(z))m γ∈Γ (defined for z outside the limit set of Γ) converges uniformly to an automorphic form of dimension m;see[63, p.218]. Poincar´e called these series “θ-fuchsian series of order m”, but the name “Poincar´e series” was later used to refer to such objects.8 The question of for which m<2 the Poincar´e series still converges was investigated by Schottky, Burnside, Fricke, and Ritter; cf. [2, pp.37-38]. In what would initially appear to be an unrelated development, mathematicians began to study the “thickness” of the limit set of a Fuchsian group: in 1941 Myrberg [135] showed that the limit set Λ of a nonelementary Fuchsian group has positive logarithmic capacity; this was improved by Beardon [17] who showed that Λ has positive Hausdorff dimension, thus deducing Myrberg’s result as a corollary (since positive Hausdorff dimension implies positive logarithmic capacity for compact sub- sets of R2 [166]). The connection between this question and the Poincar´eserieswas first observed by Akaza, who showed that if G is a Schottky group for which the Poincar´e series converges in dimension s, then the Hausdorff s-dimensional measure of Λ is zero [5, Corollary of Theorem A]. Beardon then extended Akaza’s result to finitely generated Fuchsian groups [19, Theorem 5], as well as defining the exponent of convergence (or Poincar´eexponent) δ = δG of a Fuchsian or Kleinian group to
7As is the case for many of our results, the classical proofs use compactness in a crucial way – so here “straightforwardly” means that the statements of the theorems themselves do not require modification. 8The modern definition of Poincar´e series (cf. Definition 8.1.1) is phrased in terms of hy- perbolic geometry rather than complex analysis, but it agrees with the special case of Poincar´e’s original definition which occurs when H ≡ 1andz = 0, with the caveat that γ (z)m should be replaced by |γ (z)|m. 1.2. THE BISHOP–JONES THEOREM AND ITS GENERALIZATION xxv be the infimum of s for which the Poincar´e series converges in dimension s (cf. Def- inition 8.1.1 and [18]). The reverse direction was then proven by Patterson [142] using a certain measure on Λ to produce the lower bound, which we will say more about below in §1.4. Patterson’s results were then generalized by Sullivan [161]to the setting of geometrically finite Kleinian groups. The necessity of the geometri- cal finiteness assumption was demonstrated by Patterson [143], who showed that there exist Kleinian groups of the first kind (i.e. with limit set equal to ∂Hd) with arbitrarily small Poincar´eexponent[143](seealso[100]or[157, Example 8] for an earlier example of the same phenomenon). Generalizing these theorems beyond the geometrically finite case requires the introduction of the radial and uniformly radial limit sets. In what follows, we will denote these sets by Λr and Λur, respectively. Note that the radial and uniformly radial limit sets as well as the Poincar´e exponent can all (with some care) be defined for general hyperbolic metric spaces; see Definitions 7.1.2, 7.2.1, and 8.1.1. The radial limit set was introduced by Hedlund in 1936 in his analysis of transitivity of horocycles [90, Theorem 2.4]. After some intermediate results [72, 158], Bishop and Jones [28,Theorem 1] generalized Patterson and Sullivan by proving that if G is a nonelementary 9 Kleinian group, then dimH (Λr)=dimH (Λur)=δ. Further generalization was made by Paulin [144], who proved the equation dimH (Λr)=δ inthecasewhere G ≤ Isom(X), and X is either a word-hyperbolic group, a CAT(-1) manifold, or a locally finite unweighted simplicial tree which admits a discrete cocompact action. We may now state the first major theorem of this monograph, which generalizes all the aforementioned results: Theorem 1.2.1. Let G ≤ Isom(X) be a nonelementary group. Suppose either that (1) G is strongly discrete, (2) X is a CAT(-1) space and G is moderately discrete, (3) X is an algebraic hyperbolic space and G is weakly discrete, or that (4) X is an algebraic hyperbolic space and G acts irreducibly (cf. Section 7.6) and is COT-discrete. Then there exists σ>0 such that
(1.2.1) dimH (Λr)=dimH (Λur)=dimH (Λur ∩ Λr,σ)=δ
(cf. Definitions 7.1.2 and 7.2.1 for the definition of Λr,σ); moreover, for every 10 0
9 Although Bishop and Jones’ theorem only states that dimH (Λr)=δ, they remark that their proof actually shows that dimH (Λur)=δ [28, p.4]. 10Recall that a measure μ on a metric space Z is called Ahlfors s-regular if for all z ∈ Z and 0 Remark. The “moreover” clause is new even in the case which Bishop and Jones considered, demonstrating that the limit set Λur can be approximated by subsets which are particularly well distributed from a geometric point of view. It does not follow from their theorem since a set could have large Hausdorff dimension without having any closed Ahlfors regular subsets of positive dimension (much less full dimension); in fact it follows from the work of Kleinbock and Weiss [116]that the set of well approximable numbers forms such a set.11 In [73], a slight strength- ening of this clause was used to deduce the full dimension of badly approximable vectors in the radial limit set of a Kleinian group [73, Theorem 9.3]. Remark. It is possible for a group satisfying one of the hypotheses of Theorem 1.2.1 to also satisfy δ = ∞ (Examples 13.2.1-13.3.3 and 13.5.1-13.5.2);12 note that Theorem 1.2.1 still holds in this case. Remark. A natural question is whether (1.2.2) can be improved by showing that there exists some σ>0 for which dimH (Λur,σ)=δ (cf. Definitions 7.1.2 and 7.2.1 for the definition of Λur,σ). The answer is negative. For a counterexample, 2 take X = H and G =SL2(Z) ≤ Isom(X); then for all σ>0thereexistsε>0 such that Λur,σ ⊆ BA(ε), where BA(ε) denotes the set of all real numbers with Lagrange constant at most 1/ε. (This follows e.g. from making the correspondence in [73, Observation 1.15 and Proposition 1.21] explicit.) It is well-known (see e.g. [118] for a more precise result) that dimH (BA(ε)) < 1 for all ε>0, demonstrating that dimH (Λur,σ) < 1=δ. Remark. Although Theorem 1.2.1 computes the Hausdorff dimension of the radial and uniformly radial limit sets, there are many other subsets of the limit set whose Hausdorff dimension it does not compute, such as the horospherical limit set (cf. Definitions 7.1.3 and 7.2.1) and the “linear escape” sets (Λα)α∈(0,1) [122]. We plan on discussing these issues at length in [57]. Finally, let us also remark that the hypotheses (1) - (4) cannot be weakened in any of the obvious ways. 11It could be objected that this set is not closed and therefore should not constitute a counterexample. However, since it has full measure, it has closed subsets of arbitrarily large measure (which in particular still have dimension 1). 12For the parabolic examples, take a Schottky product (Definition 10.2.1) with a lineal group (Definition 6.2.13) to get a nonelementary group, as suggested at the beginning of Chapter 13. 1.2. THE BISHOP–JONES THEOREM AND ITS GENERALIZATION xxvii Proposition 1.2.2. We may have dimH (Λr) <δeven if: (1) G is moderately discrete (even properly discontinuous) (Example 13.4.4). (2) X is a proper CAT(-1) space and G is weakly discrete (Example 13.4.1). (3) X = H∞ and G is COT-discrete (Example 13.4.9). (4) X = H∞ and G is irreducible and UOT-discrete (Example 13.4.2). (5) X = H2 (Example 13.4.5). In each case the counterexample group G is of general type (see Definition 6.2.13) and in particular is nonelementary. 1.2.1. The modified Poincar´eexponent.The examples of Proposition 1.2.2 illustrate that the Poincar´e exponent does not always accurately calculate the Hausdorff dimension of the radial and uniformly radial limit sets. In Chapter 8 we introduce a modified version of the Poincar´e exponent which succeeds at accu- rately calculating dimH (Λr) and dimH (Λur) for all nonelementary groups G.(When G is an elementary group, dimH (Λr)=dimH (Λur) = 0, so there is no need for a sophisticated calculation in this case.) Some motivation for the following definition is given in §8.2. Definition 8.2.3. Let G be a subsemigroup of Isom(X). • For each set S ⊆ X and s ≥ 0, let −sx Σs(S)= b x∈S Δ(S)={s ≥ 0:Σs(S)=∞} δ(S)=supΔ(S). • The modified Poincar´esetof G is the set (8.2.2) ΔG = Δ(Sρ), ρ>0 Sρ where the second intersection is taken over all maximal ρ-separated sets Sρ ⊆ G(o). • The number δG =supΔG is called the modified Poincar´eexponentof G. 13 If δG ∈ ΔG,wesaythatG is of generalized divergence type, while if δG ∈ [0, ∞) \ ΔG,wesaythatG is of generalized convergence type.Note that if δG = ∞,thenG is neither of generalized convergence type nor of generalized divergence type. We may now state the most powerful version of our Bishop–Jones theorem: Theorem 1.2.3 (Proven in Chapter 9). Let G Isom(X) be a nonelementary semigroup. There exists σ>0 such that (1.2.2) dimH (Λr)=dimH (Λur)=dimH (Λur ∩ Λr,σ)=δ. Moreover, for every 0 13We use the adjective “generalized” rather than “modified” because all groups of conver- gence/divergence type are also of generalized convergence/divergence type; see Corollary 8.2.8 below. xxviii 1. INTRODUCTION AND OVERVIEW Theorem 1.2.1 can be deduced as a corollary of Theorem 1.2.3; specifically, Propositions 8.2.4(ii) and 9.3.1 show that any group satisfying the hypotheses of Theorem 1.2.1 satisfies δ = δ , and hence for such a group (1.2.2) implies (1.2.1). On the other hand, Proposition 1.2.2 shows that Theorem 1.2.3 applies in many cases where Theorem 1.2.1 does not. We call a group Poincar´eregularif its Poincar´eexponentδ and modified Poincar´eexponentδ are equal. In this language, Proposition 9.3.1/Theorem 1.2.1 describes sufficient conditions for a group to be Poincar´e regular, and Proposition 1.2.2 provides a list of examples of groups which are Poincar´e irregular. Though Theorem 1.2.3 requires G to be nonelementary, the following corollary does not: Corollary 1.2.4. Fix G Isom(X). Then for some σ>0, (1.2.3) dimH (Λr)=dimH (Λur)=dimH (Λur ∩ Λr,σ). Proof. If G is nonelementary, then (1.2.3) follows from (1.2.2). On the other hand, if G is elementary, then all three terms of (1.2.3) are equal to zero. 1.3. Examples A theory of groups acting on infinite-dimensional space would not be complete without some good ways to construct examples. Techniques used in the finite- dimensional setting, such as arithmetic construction of lattices and Dehn surgery, do not work in infinite dimensions. (The impossibility of constructing lattices in Isom(H∞) as a direct limit of arithmetic lattices in Isom(Hd) is due to known lower bounds on the covolumes of such lattices which blow up as the dimension goes to infinity; see Proposition 12.2.3 below.) Nevertheless, there is a wide variety of groups acting on H∞, including many examples of actions which have no analogue in finite dimensions. 1.3.1. Schottky products. The most basic tool for constructing groups or semigroups on hyperbolic metric spaces is the theory of Schottky products. This theory was created by Schottky in 1877 when he considered the Fuchsian group generated by a finite collection of loxodromic isometries gi described by a disjoint + − 2 \ − + collection of balls Bi and Bi with the property that gi(H Bi )=Bi .Itwas extended further in 1883 by Klein’s Ping-Pong Lemma, and used effectively by Patterson [143] to construct a “pathological” example of a Kleinian group of the first kind with arbitrarily small Poincar´eexponent. We consider here a quite general formulation of Schottky products: a collection of subsemigroups of Isom(X)issaidtobeinSchottky position if open sets can be found satisfying the hypotheses of the Ping-Pong lemma whose closure is not equal to X (cf. Definition 10.2.1). This condition is sufficient to guarantee that the product of groups in Schottky position (called a Schottky product) is always COT- discrete, but stronger hypotheses are necessary in order to prove stronger forms of discreteness. There is a tension here between hypotheses which are strong enough to prove useful theorems and hypotheses which are weak enough to admit interesting examples. For the purposes of this monograph we make a fairly strong assumption (the strong separation condition, Definition 10.3.1), one which rules out infinitely generated Schottky groups whose generating regions have an accumulation point 1.3. EXAMPLES xxix (for example, infinitely generated Schottky subgroups of Isom(Hd)). However, we plan on considering weaker hypotheses in future work [57]. One theorem of significance in Chapter 10 is Theorem 10.4.7, which relates the limit set of a Schottky product to the limit set of its factors together with the image of a Cantor set ∂Γ under a certain symbolic coding map π : ∂Γ → ∂X. As a consequence, we deduce that the properties of compact type and geometri- cal finiteness are both preserved under finite strongly separated Schottky products (Corollary 10.4.8 and Proposition 12.4.19, respectively). A result analogous to The- orem 10.4.7 in the setting of infinite alphabet conformal iterated function systems can be found in [128, Lemma 2.1]. In §10.5, we discuss some (relatively) explicit constructions of Schottky groups, showing that Schottky products are fairly ubiquitous - for example, any two groups which act properly discontinuously at some point of ∂X may be rearranged to be in Schottky position, assuming that X is sufficiently symmetric (Proposition 10.5.1). 1.3.2. Parabolic groups. A major point of departure where the theory of subgroups of Isom(H∞) becomes significantly different from the finite-dimensional theory is in the study of parabolic groups. As a first example, knowing that a group admits a discrete parabolic action on Isom(X) places strong restrictions on the al- d ∞ gebraic properties of the group if X = HF, but not if X = HF . Concretely, discrete d parabolic subgroups of Isom(HF) are always virtually nilpotent (virtually abelian if F = R), but any group with the Haagerup property admits a parabolic strongly discrete action on H∞ (indeed, this is a reformulation of one of the equivalent def- initions of the Haagerup property; cf. [50, p.1, (4)]). Examples of groups with the Haagerup property include all amenable groups and free groups. Moreover, strongly discrete parabolic subgroups of Isom(H∞) need not be finitely generated; cf. Example 11.2.20. Moving to infinite dimensions changes not only the algebraic but also the geo- metric properties of parabolic groups. For example, the cyclic group generated by a parabolic isometry may fail to be discrete in any reasonable sense (Example 11.1.12), or it may be discrete in some senses but not others (Example 11.1.14). d The Poincar´e exponent of a parabolic subgroup of Isom(HF)isalwaysahalf-integer [54, Proof of Lemma 3.5], but the situation is much more complicated in infinite dimensions. We prove a general lower bound on the Poincar´eexponentofapara- bolic subgroup of Isom(X) for any hyperbolic metric space X, depending only on the algebraic structure of the group (Theorem 11.2.6); in particular, the Poincar´e exponent of a parabolic action of Zk on a hyperbolic metric space is always at least k/2. Of course, it is well-known that all parabolic actions of Zk on Hd achieve equality. By contrast, we show that for every δ>k/2 there exists a parabolic action of Zk on H∞ whose Poincar´eexponentisequaltoδ (Theorem 11.2.11). 1.3.3. Geometrically finite and convex-cobounded groups. It has been known for a long time that every finitely generated Fuchsian group has a finite-sided convex fundamental domain (e.g. [108, Theorem 4.6.1]). This result does not gen- eralize beyond two dimensions (e.g. [25, 102]), but subgroups of Isom(H3) with finite-sided fundamental domains came to be known as geometrically finite groups. Several equivalent definitions of geometrical finiteness in the three-dimensional set- ting became known, for example Beardon and Maskit’s condition that the limit set is the union of the radial limit set Λr with the set Λbp of bounded parabolic points xxx 1. INTRODUCTION AND OVERVIEW [21], but the situation in higher dimensions was somewhat murky until Bowditch [34] wrote a paper which described which equivalences remain true in higher di- mensions, and which do not. The condition of a finite-sided convex fundamental domain is no longer equivalent to any other conditions in higher dimensions (e.g. [12]), so a higher-dimensional Kleinian group is said to be geometrically finite if it satisfies any of Bowditch’s five equivalent conditions (GF1)-(GF5). In infinite dimensions, conditions (GF3)-(GF5) are no longer useful (cf. Remark 12.4.6), but appropriate generalizations of conditions (GF1) (convex core is equal to a compact set minus a finite number of cusp regions) and (GF2) (the Beardon– Maskit formula Λ = Λr∪Λbp) are still equivalent for groups of compact type. In fact, (GF1) is equivalent to (GF2) + compact type (Theorem 12.4.5). We define a group to be geometrically finite if it satisfies the appropriate analogue of (GF1) (Definition 12.4.1). A large class of examples of geometrically finite subgroups of Isom(H∞) is furnished by combining the techniques of Chapters 10 and 11; specifically, the strongly separated Schottky product of any finite collection of parabolic groups and/or cyclic loxodromic groups is geometrically finite (Corollary 12.4.20). It remains to answer the question of what can be proven about geometrically finite groups. This is a quite broad question, and in this monograph we content ourselves with proving two theorems. The first theorem, Theorem 12.4.14, is a generalization of the Milnor–Schwarz lemma [39, Proposition I.8.19] (see also The- orem 12.2.12), and describes both the algebra and geometry of a geometrically finite group G:firstly,G is generated by a finite subset F ⊆ G together with a finite collection of parabolic subgroups Gξ (which are not necessarily finitely generated, e.g. Example 11.2.20), and secondly, the orbit map g → g(o) is a quasi-isometric embedding from (G, dG)intoX,wheredG is a certain weighted Cayley metric (cf. ∪ Example 3.1.2 and (12.4.6)) on G whose generating set is F ξ Gξ. As a conse- quence (Corollary 12.4.17), we see that if the groups Gξ, ξ ∈ Λbp, are all finitely generated, then G is finitely generated, and if these groups have finite Poincar´e exponent, then G has finite Poincar´eexponent. 1.3.4. Counterexamples. A significant class of subgroups of Isom(H∞)that has no finite-dimensional analogue is provided by the Burger–Iozzi–Monod (BIM) representation theorem [40, Theorem 1.1], which states that any unweighted sim- plicial tree can be equivariantly and quasi-isometrically embedded into an infinite- dimensional real hyperbolic space, with a precise relation between distances in the domain and distances in the range. We call the embeddings provided by their theorem BIM embeddings, and the corresponding homomorphisms provided by the equivariance we call BIM representations. We generalize the BIM embedding theo- rem to the case where X is a separable R-tree rather than an unweighted simplicial tree (Theorem 13.1.1). If we have an example of an R-tree X and a subgroup Γ ≤ Isom(X)witha certain property, then the image of Γ under a BIM representation generally has the same property (Remark 13.1.4). Thus, the BIM embedding theorem allows us to translate counterexamples in R-trees into counterexamples in H∞.Forex- ample, if Γ is the free group on two elements acting on its Cayley graph, then the image of Γ under a BIM representation provides a counterexample both to an infinite-dimensional analogue of Margulis’s lemma (cf. Example 13.1.5) and to an infinite-dimensional analogue of I. Kim’s theorem regarding length spectra of finite-dimensional algebraic hyperbolic spaces (cf. Remark 13.1.6). 1.4. PATTERSON–SULLIVAN THEORY xxxi Most of the other examples in Chapter 13 are concerned with our various notions of discreteness (cf. §1.1.3 above), the notion of Poincar´e regularity (i.e. whetherornotδ = δ ), and the relations between them. Specifically, we show that the only relations are the relations which were proven in Chapter 5 and Proposi- tion 9.3.1, as summarized in Table 1, p.77. Perhaps the most interesting of the counterexamples we give is Example 13.4.2, which is the image under a BIM rep- resentation of (a countable dense subgroup of) the automorphism group Γ of the 4-regular unweighted simplicial tree. This example is notable because discreteness properties are not preserved under taking the BIM representation: specifically, Γ is weakly discrete but its image under the BIM representation is not. It is also interesting to try to visualize this image geometrically (cf. Figure 13.4.1). 1.3.5. R-trees and their isometry groups. Motivated by the BIM rep- resentation theorem, we discuss some ways of constructing R-trees which admit natural isometric actions. Our first method is the cone construction, in which one starts with an ultrametric space (Z, D) and builds an R-tree X asa“cone”over Z. This construction first appeared in a paper of F. Choucroun [52], although it is similar to several other known cone constructions: [85, 1.8.A.(b)], [168], [31, §7]. R-trees constructed by the cone method tend to admit natural parabolic actions, and in Theorem 14.1.5 we provide a necessary and sufficient condition for a function to be the orbital counting function of some parabolic group acting on an R-tree. Our second method is to staple R-trees together to form a new R-tree. We give sufficient conditions on a graph (V,E), a collection of R-trees (Xv)v∈V ,andacol- lection of sets A(v, w) ⊆ Xv and bijections ψv,w : A(v, w) → A(w, v)((v, w) ∈ E) such that stapling the trees (Xv)v∈V along the isometries (ψv,w)(v,w)∈E yields an R-tree (Theorem 14.4.4). In §14.5, we give three examples of the stapling construc- tion, including looking at the cone construction as a special case of the stapling construction. The stapling construction is somewhat similar to a construction of G. Levitt [120]. 1.4. Patterson–Sullivan theory The connection between the Poincar´eexponentδ of a Kleinian group and the geometry of its limit set is not limited to Hausdorff dimension considerations such as those in the Bishop–Jones theorem. As we mentioned before, Patterson and Sullivan’s proofs of the equality dimH (Λ) = δ for geometrically finite groups rely on the construction of a certain measure on Λ, the Patterson–Sullivan measure,whose Hausdorff dimension is also equal to δ. In addition to connecting the Poincar´e exponent and Hausdorff dimension, the Patterson–Sullivan measure also relates to the spectral theory of the Laplacian (e.g. [142, Theorem 3.1], [161, Proposition 28]) and the geodesic flow on the quotient manifold [103]. An important property of Patterson–Sullivan measures is conformality.Givens>0, a measure μ on ∂Bd is said to be s-conformal with respect to a discrete group G ≤ Isom(Bd)if (1.4.1) μ(g(A)) = |g(ξ)|s dμ(ξ) ∀g ∈ G ∀A ⊆ ∂Bd. A The Patterson–Sullivan theorem on the existence of conformal measures may now be stated as follows: For every Kleinian group G, there exists a δ-conformal measure on Λ, where δ is the Poincar´eexponentofG and Λ is the limit set of G. xxxii 1. INTRODUCTION AND OVERVIEW When dealing with “coarse” spaces such as arbitrary hyperbolic metric spaces, it is unreasonable to expect equality in (1.4.1). Thus, a measure μ on ∂X is said to be s-quasiconformal with respect to a group G ≤ Isom(X)if s μ(g(A)) × g (ξ) dμ(ξ) ∀g ∈ G ∀A ⊆ ∂X. A Here g(ξ) denotes the upper metric derivative of g at ξ;cf.§4.2.2. We remark that if X is a CAT(-1) space and G is countable, then every quasiconformal measure is coarsely asymptotic to a conformal measure (Proposition 15.2.1). In Chapter 15, we describe the theory of conformal and quasiconformal mea- sures in hyperbolic metric spaces. The main theorem is the existence of δ -conformal measures for groups of compact type (Theorem 15.4.6). An important special case of this theorem has been proven by Coornaert [53,Th´eor`eme 5.4] (see also [41, §1], [152, Lemme 2.1.1]): the case where X is proper and geodesic and G satisfies δ<∞. The main improvement from Coornaert’s theorem to ours is the ability to construct quasiconformal measures for Poincar´e irregular (δ<δ = ∞) groups; this improvement requires an argument using the class of uniformly continuous functions on bord X. The big assumption of Theorem 15.4.6 is the assumption of compact type. All proofs of the Patterson–Sullivan theorem seem to involve taking a weak-* limit of a sequence of measures in X and then proving that the limit measure is (quasi)conformal, but how can we take a weak-* limit if the limit set is not compact? In fact, Theorem 15.4.6 becomes false if you remove the assumption of compact type. In Proposition 16.6.1, we construct a group acting on an R-tree and satisfying δ<∞ which admits no δ-conformal measure on its limit set, and then use the BIM embedding theorem (Theorem 13.1.1) to get an example in H∞. Surprisingly, it turns out that if we replace the hypothesis of compact type with the hypothesis of divergence type, then the theorem becomes true again. Specifically, we have the following: Theorem 1.4.1 (Proven in Chapter 16). Let G ≤ Isom(X) be a nonelementary group of generalized divergence type (see Definition 8.2.3). Then there exists a δ - quasiconformal measure μ for G supported on Λ,whereδ is the modified Poincar´e exponent of G. It is unique up to a multiplicative constant in the sense that if μ1,μ2 are two such measures then μ1 × μ2 (cf. Remark 15.1.2). In addition, μ is ergodic and gives full measure to the radial limit set of G. To motivate Theorem 1.4.1, we recall the connection between the divergence type condition and Patterson–Sullivan theory in finite dimensions. Although the Patterson–Sullivan theorem guarantees the existence of a δ-conformal measure, it does not guarantee its uniqueness. Indeed, the δ-conformal measure is often not unique; see e.g. [10]. However, it turns out that the hypothesis of divergence type is enough to guarantee uniqueness. In fact, the condition of divergence type turns out to be quite important in the theory of conformal measures: Theorem 1.4.2 (Hopf–Tsuji–Sullivan theorem, [138, Theorem 8.3.5]). Fix d ≥ 2,letG ≤ Isom(Hd) be a discrete group, and let δ be the Poincar´eexponentofG. Then for any δ-conformal measure μ ∈M(Λ), the following are equivalent: (A) G is of divergence type. (B) μ gives full measure to the radial limit set Λr(G). 1.4. PATTERSON–SULLIVAN THEORY xxxiii (C) G acts ergodically on (Λ,μ) × (Λ,μ). In particular, if G is of divergence type, then every δ-conformal measure is ergodic, so there is exactly one (ergodic) δ-conformal probability measure. We remark that our sentence “In particular . . . ” stated in theorem above was not included in [138, Theorem 8.3.5] but it is well-known and follows easily from the equivalence of (A) and (C). Remark 1.4.3. Theorem 1.4.2 has a long history. The equivalence (B) ⇔ (C) was first proven by E. Hopf in the case δ = d − 114 [99, 100] (1936, 1939). The equivalence (A) ⇔ (B) was proven by Z. Yˆujˆobˆointhecaseδ = d − 1=1[176] (1949), following an incorrect proof by M. Tsuji [169] (1944).15 Sullivan proved (A) ⇔ (C) in the case δ = d − 1[163, Theorem II], then generalized this equivalence to the case δ>(d − 1)/2[161, Theorem 32]. He also proved (B) ⇔ (C) in full generality [161, Theorem 21]. Next, W. P. Thurston gave a simpler proof of (A) ⇒ (B)16 in the case δ = d − 1[4, Theorem 4 of Section VII]. P. J. Nicholls finished the proof by showing (A) ⇔ (B) in full generality [138, Theorems 8.2.2 and 8.2.3]. Later S. Hong re-proved (A) ⇒ (B) in full generality twice in two independent papers [97, 98], apparently unaware of any previous results. Another proof of (A) ⇒ (B) in full generality, which was conceptually similar to Thurston’s proof, was given by P. Tukia [171, Theorem 3A]. Further generalization was made by C. Yue [175] to negatively curved manifolds, and by T. Roblin [151,Th´eor`eme 1.7] to proper CAT(-1) spaces. Having stated the Hopf–Tsuji–Sullivan theorem, we can now describe why The- orem 1.4.1 is true, first on an intuitive level and then giving a sketch of the real proof. On an intuitive level, the fact that divergence type implies both “existence and uniqueness” of the δ-conformal measure in finite dimensions indicates that per- haps the compactness assumption is not needed – the sequence of measures used to construct the Patterson–Sullivan measure converges already, so it should not be necessary to use compactness to take a convergent subsequence. The real proof involves taking the Samuel–Smirnov compactification of bord X, considered as a metric space with respect to a visual metric (cf. §3.6.3). The Samuel–Smirnov compactification of a metric space (cf. [136, §7]) is conceptually similar to the more familiar Stone–Cechˇ compactification, except that only uni- formly continuous functions on the metric space extend to continuous functions on the compactification, not all continuous functions. If we used the Stone–Cechˇ compactification rather than the Samuel–Smirnov compactification, then our proof would only apply to groups with finite Poincar´e exponent; cf. Remark 16.1.3 and Remark 16.3.5. Sketch of the proof of Theorem 1.4.1. We denote the Samuel–Smirnov compactification of bord X by X . By a nonstandard analogue of Theorem 15.4.6 (viz. Lemma 16.3.4), there exists a δ -quasiconformal measure μ on ∂X .Bya generalization of Theorem 1.4.2 (viz. Proposition 16.4.1), μ gives full measure to 14In this paragraph, when we say that someone proves the case δ = d − 1, we mean that they considered the case where μ is Hausdorff (d − 1)-dimensional measure on Sd−1. 15See [163, p.484] for some further historical remarks on the case δ = d − 1=1. 16By this point, it was considered obvious that (B) ⇒ (A). xxxiv 1. INTRODUCTION AND OVERVIEW the radial limit set Λr. But a simple computation (Lemma 16.2.5) shows that Λr =Λr, demonstrating that μ ∈M(Λ). 1.4.1. Quasiconformal measures of geometrically finite groups. Let us consider a geometrically finite group G ≤ Isom(X) with Poincar´eexponent δ<∞, and let μ be a δ-quasiconformal measure on Λ. Such a measure exists since geometrically finite groups are of compact type (Theorem 12.4.5 and Theorem 15.4.6), and is unique as long as G is of divergence type (Corollary 16.4.6). When X = Hd, the geometry of μ is described by the Global Measure Formula [165, Theorem on p.271], [160, Theorem 2]: the measure of a ball B(η, e−t)iscoarsely −δt asymptotic to e times a factor depending on the location of the point ηt := [o, η]t d in the quotient manifold H /G.Here[o, η]t is the unique point on the geodesic connecting o and η with distance t from o; cf. Notations 3.1.6, 4.4.3. In a general hyperbolic metric space X (indeed, already for X = H∞), one cannot get a precise asymptotic for μ(B(η, e−t)), due to the fact that the measure μ may fail to be doubling (Example 17.4.12). Instead, our version of the global measure formula gives both an upper bound and a lower bound for μ(B(η, e−t)). Specifically, we define a function m :Λ× [0, ∞) → (0, ∞) (for details see (17.2.1)) and then show: Theorem 1.4.4 (Global measure formula, Theorem 17.2.2; proven in Section 17.3). For all η ∈ Λ and t>0, −t (1.4.2) m(η, t + σ) × μ(B(η, e )) × m(η, t − σ), where σ>0 is independent of η and t. It is natural to ask for which groups (1.4.2) can be improved to an exact as- ymptotic, i.e. for which groups μ is doubling. We address this question in Section 17.4, proving a general result (Proposition 17.4.8), a special case of which is that if X is a finite-dimensional algebraic hyperbolic space, then μ is doubling (Example 17.4.11). Nevertheless, there are large classes of examples of groups G ≤ Isom(H∞) for which μ is not doubling (Example 17.4.12), illustrating once more the wide difference between H∞ and its finite-dimensional counterparts. It is also natural to ask about the implications of the Global Measure Formula for the dimension theory of the measure μ. For example, when X = Hd, the Global Measure Formula was used to show that dimH (μ)=δ [160, Proposition 4.10]. In our case we have: Theorem 1.4.5 (Cf. Theorem 17.5.9). If for all p ∈ P ,theseries (1.4.3) e−δh h h∈Gp converges, then μ is exact dimensional (cf. Definition 17.5.2)ofdimensionδ.In particular, dimH (μ)=dimP (μ)=δ. The hypothesis that (1.4.3) converges is a very non-restrictive hypothesis. For example, it is satisfied whenever δ>δp for all p ∈ P (Corollary 17.5.10). Combining with Proposition 10.3.10 shows that any counterexample must satisfy e−δh < ∞ = e−δh h h∈Gp h∈Gp 1.5. APPENDICES xxxv for some p ∈ P , creating a very narrow window for the orbital counting function Np (cf. Notation 17.2.1) to lie in. Nevertheless, we show that there exist coun- terexamples (Example 17.5.14) for which the series (1.4.3) diverges. After making some simplifying assumptions, we are able to prove (Theorem 17.5.13) that the Patterson–Sullivan measures of groups for which (1.4.3) diverges cannot be exact dimensional, and in fact satisfy dimH (μ)=0. There is a relation between exact dimensionality of the Patterson–Sullivan mea- sure and the theory of Diophantine approximation on the boundary of ∂X,asde- scribed in [73]. Specifically, if VWAξ denotes the set of points which are very well approximable with respect to a distinguished point ξ (cf. §17.5.1), then we have the following: Theorem 1.4.6 (Cf. Theorem 17.5.8). The following are equivalent: (A) For all p ∈ P , μ(VWAp)=0. (B) μ is exact dimensional. (C) dimH (μ)=δ. (D) For all ξ ∈ Λ, μ(VWAξ)=0. In particular, combining with Theorem 1.4.5 demonstrates that the equation μ(VWAξ)=0 holds for a large class of geometrically finite groups G and for all ξ ∈ Λ. This improves the results of [73, §1.5.3]. 1.5. Appendices We conclude this monograph with two appendices. 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Soc. 348 (1996), no. 12, 4965–5005. 176. Z. Yˆujˆobˆo, A theorem on Fuchsian groups, Math. Japonicae 1 (1949), 168–169. Selected Published Titles in This Series 218 Tushar Das, David Simmons, and Mariusz Urba´nski, Geometry and Dynamics in Gromov Hyperbolic Metric Spaces, 2017 215 Robert Bieri and Ralph Strebel, On Groups of PL-homeomorphisms of the Real Line, 2016 214 Jared Speck, Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations, 2016 213 Harold G. Diamond and Wen-Bin Zhang (Cheung Man Ping), Beurling Generalized Numbers, 2016 212 Pandelis Dodos and Vassilis Kanellopoulos, Ramsey Theory for Product Spaces, 2016 211 Charlotte Hardouin, Jacques Sauloy, and Michael F. Singer, Galois Theories of Linear Difference Equations: An Introduction, 2016 210 Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker, The Dynamical Mordell–Lang Conjecture, 2016 209 Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, 2015 208 Peter S. Ozsv´ath, Andr´as I. Stipsicz, and Zolt´an Szab´o, Grid Homology for Knots and Links, 2015 207 Vladimir I. Bogachev, Nicolai V. Krylov, Michael R¨ockner, and Stanislav V. Shaposhnikov, Fokker–Planck–Kolmogorov Equations, 2015 206 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow: Techniques and Applications: Part IV: Long-Time Solutions and Related Topics, 2015 205 Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, 2015 204 Victor M. Buchstaber and Taras E. Panov, Toric Topology, 2015 203 Donald Yau and Mark W. Johnson, A Foundation for PROPs, Algebras, and Modules, 2015 202 Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, Asymptotic Geometric Analysis, Part I, 2015 201 Christopher L. Douglas, John Francis, Andr´e G. Henriques, and Michael A. Hill, Editors, Topological Modular Forms, 2014 200 Nikolai Nadirashvili, Vladimir Tkachev, and Serge Vl˘adut¸, Nonlinear Elliptic Equations and Nonassociative Algebras, 2014 199 Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of Free Noncommutative Function Theory, 2014 198 J¨org Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, 2014 197 Richard Evan Schwartz, The Octagonal PETs, 2014 196 Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of Isotropic Convex Bodies, 2014 195 Ching-Li Chai, Brian Conrad, and Frans Oort, Complex Multiplication and Lifting Problems, 2014 194 Samuel Herrmann, Peter Imkeller, Ilya Pavlyukevich, and Dierk Peithmann, Stochastic Resonance, 2014 193 Robert Rumely, Capacity Theory with Local Rationality, 2013 192 Messoud Efendiev, Attractors for Degenerate Parabolic Type Equations, 2013 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/. This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit Credit: Christine Cockett sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems. For additional information and updates on this book, visit AMS on the Web www.ams.org/bookpages/surv-218 www.ams.org SURV/218